Asymptotic Symmetries in $p$-Form Theories

TL;DR

The paper investigates asymptotic symmetries of $(d,p)$-form gauge theories in flat spacetime, identifying a critical dimension $d=2p+4$ where radiation and Coulomb fall-offs align and finite charges arise. By formulating a Maxwell-type action with a boundary term in Lorenz gauge and employing de Sitter slicing, the authors derive conserved surface charges associated with residual gauge transformations and compute their algebra. They classify charges into zero-mode, coexact, and exact sectors; the first two are Abelian, while the exact sector for $p eq 0$ yields a non-Abelian structure with a central extension, exemplified explicitly in the $6d$ (2-form) case. The work provides concrete realizations in $4d$ Maxwell theory and $6d$ 2-form theory, including antipodal matching and mode expansions, and discusses physical implications such as memory effects and potential links to flat-space holography. Overall, the results reveal a rich higher-form asymptotic symmetry structure with distinct algebraic properties across charge sectors and dimensions, offering a platform for exploring generalized global symmetries and memory phenomena in higher-form gauge theories.

Abstract

We consider $(p+1)$-form gauge fields in flat $(2p+4)$-dimensions for which the radiation and the Coulomb solutions have the same asymptotic falloff behavior. Imposing appropriate falloff behavior on fields and adopting a Maxwell-type action, we construct the boundary term which renders the action principle well-defined in the Lorenz gauge. We then compute conserved surface charges and the corresponding asymptotic charge algebra associated with nontrivial gauge transformations. We show that for $p\geq 1$ cases we have three sets of conserved asymptotic charges associated with exact, coexact and zero-mode parts of the corresponding $p$-form gauge transformations on the asymptotic $S^{2p+2}$. The coexact and zero-mode charges are higher form extensions of the four dimensional electrodynamics case $(p=0)$, and are commuting. Charges associated with exact gauge transformations have no counterparts in four dimensions and form infinite copies of Heisenberg algebras. We briefly discuss physical implications of these charges and their algebra.

Figures (4)

• Figure 1: Penrose diagrams of Minkowski flat spacetime ${\cal M}_d$. (Left) The dashed and thick curves denote the constant Cartesian time $t$ and radial $r$ slices respectively. As we see all the constant $t$ curves meet the asymptotic spatial infinity $i_0$ with a zero slope. (Right) The patch covered by the de Sitter slicing. The solid lines are constant $\tau$ slices, while dotted lines are constant $\rho$ hyperboloids. The de Sitter slicing does not cover future and past timelike infinities $i^\pm$.
• Figure 2: Embedding the de sitter space in global patch as $\rho$-constant slices of Minkowski pace with $\rho^2=x_\mu x^\mu$. The $\rho_0\to\infty$ region gives the $dS_{d-1}$ at the boundary of the Minkowski space.
• Figure 3: $I_1, I_2$ depict constant de Sitter time ($\tau=const$) and blue-shaded region $B$ shows constant $\rho$ slice of the Mink$_d$ between the two constant $\tau$ regions. $C_1,C_2$ are boundaries of these constant time slices; these are two $S^{d-2}$ spheres corresponding to codimension 2 $\rho,\tau$-constant surfaces in the Mink$_d$ region.
• Figure 4: Penrose diagram of de Sitter space. The shaded region is the causal future of the north pole, and its dashed $45^{\circ}$ boundary is the future horizon of the south pole. The antipodal map amounts to a couple of horizontal and vertical flips. Especially, the north pole at far past is mapped to the south pole at far future.